Sample Pairs

Energy Sector Pairs: XOM/CVX

This case study examines the performance of a statistical arbitrage strategy using Exxon Mobil (XOM) and Chevron (CVX), two major integrated oil and gas companies with similar business models and market exposures.

Stock Price Trend Comparison

The chart above shows the price movements of XOM (blue line) and CVX (orange line) over our analysis period. Despite the different price scales, the two stocks clearly display similar directional movements, providing the foundation for our pairs trading strategy.

Fundamental Relationship

Exxon Mobil and Chevron represent an ideal pairing for statistical arbitrage due to their strong fundamental relationship:

• Both are integrated oil majors with similar business models spanning exploration, production, refining, and retail
• Shared exposure to global oil and gas prices as primary profit drivers
• Comparable market capitalization and financial strength profiles
• Similar dividend policies and capital expenditure patterns
• Competing operations across the same geographical regions

This fundamental similarity creates a strong economic rationale for their prices to maintain a consistent relationship over time, despite short-term divergences.

Statistical Validation

Our comprehensive statistical analysis confirms the strong relationship between XOM and CVX:

Metric	Value	Interpretation
Pearson Correlation	0.8497	Strong positive correlation
Correlation Std Dev	0.3172	Moderate variability in correlation
Stability Score	-21.7160	Correlation shows some instability

Rolling Correlation

The rolling correlation chart above demonstrates the generally strong relationship between XOM and CVX over time. The 60-day rolling correlation (purple line) and 120-day rolling correlation (orange line) both remain mostly above 0.8, though there are periods of significant drops that contribute to the negative stability score.

While the correlation stability shows some concerns, the cointegration tests provide strong evidence for a tradable relationship:

Cointegration Test Results

Test	Statistic	Critical Value (5%)	p-value	Result
ADF Test	-2.9819	-2.8654	0.0366	Cointegrated ✓
Johansen Trace	24.4420	15.4943	-	Cointegrated ✓

Both tests confirm the presence of cointegration, indicating that while XOM and CVX prices may diverge temporarily, they tend to revert to their long-term equilibrium relationship. The hedge ratio of 0.1816 suggests that for each share of XOM, approximately 0.18 shares of CVX should be traded in the opposite direction to create a market-neutral position.

Spread Mean-Reverting Series

The spread chart above shows the difference between the two stocks (after applying the hedge ratio), demonstrating the mean-reverting property that is essential for pairs trading. While the spread appears non-stationary at times, our statistical tests confirm it is indeed mean-reverting over the analysis period.

Trading Parameters

Based on our statistical analysis, we established the following trading parameters:

• Entry Signal: Z-score exceeding +/- 2.0 standard deviations
• Exit Signal: Z-score returning to +/- 0.5 standard deviations
• Stop-Loss: Z-score exceeding +/- 3.0 standard deviations
• Half-Life: 42.41 days (relatively long)
• Maximum Holding Period: 63 days (1.5× half-life)

Interactive Z-Score Strategy Explorer

The interactive tool below allows you to experiment with different trading parameters and explore how they would affect the signals generated for the XOM/CVX pair. Adjust the thresholds to see how changes in your trading rules would impact potential entry and exit points.

zscore_data = FileAttachment("resources/zscore.csv").csv(d => ({
  timestamp: new Date(d.timestamp),
  XOM_price: +d.XOM_close,
  CVX_price: +d.CVX_close,
  zscore: +d.zscore
}))

// 滑动条输入：选取数据的起始和结束索引
viewof rangeStart = Inputs.range([0, zscore_data.length - 1], {
  value: 0,
  step: 1,
  label: "Start Date Index"
})

viewof rangeEnd = Inputs.range([0, zscore_data.length - 1], {
  value: 150,
  step: 1,
  label: "End Date Index"
})

// 滑动条输入：进场和离场信号的 Z-Score 阈值
viewof entryThreshold = Inputs.range(
  [1, 3],
  {value: 2, step: 0.1, label: "Entry Threshold (Z-Score):"}
)

viewof exitThreshold = Inputs.range(
  [0, 1],
  {value: 1, step: 0.1, label: "Exit Threshold (Z-Score):"}
)

// 选择需要显示的信号和股票价格曲线
viewof displayOptions = Inputs.checkbox(
  ["Entry Signals", "Exit Signals"],
  {value: ["Entry Signals", "Exit Signals"], label: "Display:"}
)

// 筛选指定区间内的数据
selectedData = zscore_data.slice(rangeStart, rangeEnd)

// 筛选进场（long/short）和离场信号
entryLongSignals = selectedData.filter(d => d.zscore <= -entryThreshold)
entryShortSignals = selectedData.filter(d => d.zscore >= entryThreshold)
exitSignals = selectedData.filter(d => {
  return Math.abs(d.zscore) <= exitThreshold && Math.abs(d.zscore) > 0.1;
})

// 绘制交互式图表
Plot.plot({
  width: 800,
  height: 500,
  marginLeft: 50,
  marginRight: 20,
  marginBottom: 40,
  style: {
    backgroundColor: "#111",  // 黑底
    color: "#eee",            // 白字
    fontSize: 12,
    fontFamily: "sans-serif"
  },
  marks: [
    // Z-Score 折线图
    Plot.line(selectedData, {
      x: "timestamp",
      y: "zscore",
      stroke: "steelblue",
      strokeWidth: 2
    }),

    // 进场信号：做多信号
    displayOptions.includes("Entry Signals") ?
      Plot.dot(entryLongSignals, {
        x: "timestamp",
        y: "zscore",
        fill: "green",
        r: 6,
        title: d => `Long Entry: ${d.timestamp} (Z: ${d.zscore})`
      }) : null,

    // 进场信号：做空信号
    displayOptions.includes("Entry Signals") ?
      Plot.dot(entryShortSignals, {
        x: "timestamp",
        y: "zscore",
        fill: "red",
        r: 6,
        title: d => `Short Entry: ${d.timestamp} (Z: ${d.zscore})`
      }) : null,

    // 离场信号
    displayOptions.includes("Exit Signals") ?
      Plot.dot(exitSignals, {
        x: "timestamp",
        y: "zscore",
        fill: "orange",
        r: 4,
        title: d => `Exit: ${d.timestamp} (Z: ${d.zscore})`
      }) : null,

    // 图中各个辅助参考线
    Plot.ruleY([0], {stroke: "gray", strokeWidth: 1}),
    Plot.ruleY([entryThreshold], {stroke: "red", strokeDasharray: "4 4"}),
    Plot.ruleY([-entryThreshold], {stroke: "green", strokeDasharray: "4 4"}),
    Plot.ruleY([exitThreshold], {stroke: "orange", strokeDasharray: "4 4", strokeOpacity: 0.7}),
    Plot.ruleY([-exitThreshold], {stroke: "orange", strokeDasharray: "4 4", strokeOpacity: 0.7})
  ],

  grid: true,
  x: {
    label: "Date →",
    type: "time",  // 明确指定这是时间类型
    tickFormat: "%Y-%m-%d",  // 直接使用格式字符串
    labelAnchor: "center"
  },
  y: {
    label: "← Z-Score →",
    domain: [-4, 4],
    labelAnchor: "center"
  },
  title: "Interactive Z-Score Analysis for XOM/CVX Pair"
})

longEntrySignals = entryLongSignals.length
shortEntrySignals = entryShortSignals.length
exitSignalsNum = exitSignals.length
avgZscoreAbs = d3.mean(selectedData, d => Math.abs(d.zscore))
pctTimeAboveThreshold = (((entryLongSignals.length + entryShortSignals.length) / selectedData.length) * 100).toFixed(1) + "%"
signalRatio = ((entryLongSignals.length + entryShortSignals.length) / Math.max(1, exitSignals.length))

// 显示指标数据表
Inputs.table(
  [
    {metric: "Long Entry Signals (Z < 0)", value: longEntrySignals},
    {metric: "Short Entry Signals (Z > 0)", value: shortEntrySignals},
    {metric: "Exit Signals", value: exitSignalsNum},
    {metric: "Average |Z-Score|", value: avgZscoreAbs},
    {metric: "% Time Beyond Entry Threshold", value: pctTimeAboveThreshold},
    {metric: "Entry/Exit Signal Ratio", value: signalRatio}
  ],
  {
    header: {
      metric: "Trading Metric",
      value: "Value"
    },
    width: {
      metric: 200,
      value: 100
    }
  }
)

The half-life analysis indicates that while mean reversion does occur, it happens at a relatively slow pace. This suggests a medium-term trading horizon for this pair.

Z-Score with Trading Signals

The Z-score chart above shows the normalized spread with our entry and exit thresholds. When the Z-score crosses above +2 or below -2 (red dashed lines), we enter a trade in the direction of mean reversion. The frequent oscillation of the Z-score around these thresholds provides numerous trading opportunities.

Trading Performance

Our backtest of the XOM/CVX pair trading strategy demonstrates impressive results over the test period:

Equity Curve

The equity curve above shows the growth of our initial $10,000 capital over the testing period. The steady upward trend demonstrates the consistent profitability of the strategy, reaching over $13,600 by the end of the period.

Detailed Trade Analysis

To provide deeper insights into our trading activity, we analyzed the complete trade blotter and ledger results from our backtest:

Timestamp	Timestamp.1	XOM Position	CVX Position	Quantity	XOM Entry	XOM Exit	XOM P&L	CVX Entry	CVX Exit	CVX P&L	Total P&L	success
2022-07-27	2022-08-03	Short	Long	54	$91.57	$91.02	$29.70	$149.26	$155.36	$36.60	$66.30	True
2022-08-23	2022-09-01	Short	Long	51	$98.00	$93.87	$210.63	$161.99	$155.54	$-38.70	$171.93	True
2022-09-20	2022-10-03	Long	Short	55	$92.44	$91.92	$-28.60	$156.28	$151.73	$27.30	$-1.30	False
2022-10-05	2022-11-09	Short	Long	51	$99.12	$108.90	$-498.78	$158.53	$177.93	$116.40	$-382.38	False
2022-12-05	2022-12-13	Long	Short	46	$106.85	$107.25	$18.40	$176.56	$173.53	$15.15	$33.55	True
2023-01-06	2023-02-02	Short	Long	44	$110.53	$111.15	$-27.28	$176.56	$169.01	$-37.75	$-65.03	False
2023-02-10	2023-02-17	Short	Long	41	$119.17	$111.28	$323.49	$171.97	$162.85	$-45.60	$277.89	True
2023-03-13	2023-03-21	Long	Short	47	$106.54	$107.04	$23.50	$158.71	$159.31	$-3.60	$19.90	True
2023-04-03	2023-05-01	Short	Long	43	$116.13	$114.67	$62.78	$169.95	$167.24	$-13.55	$49.23	True
2023-05-02	2023-05-24	Long	Short	46	$110.10	$107.59	$-115.46	$160.04	$157.27	$16.62	$-98.84	False
2023-05-31	2023-06-02	Long	Short	49	$102.18	$105.76	$175.42	$150.62	$156.26	$-33.84	$141.58	True
2023-06-07	2023-06-12	Short	Long	47	$108.53	$106.42	$99.17	$159.83	$157.33	$-15.00	$84.17	True
2023-06-20	2023-06-26	Long	Short	50	$102.72	$104.29	$78.50	$153.68	$154.01	$-1.98	$76.52	True
2023-07-14	2023-07-24	Long	Short	51	$100.94	$105.52	$233.58	$153.58	$161.82	$-49.44	$184.14	True
2023-08-09	2023-08-16	Short	Long	48	$109.56	$106.34	$154.56	$159.86	$158.63	$-7.38	$147.18	True
2023-09-01	2023-09-21	Short	Long	47	$113.52	$114.76	$-58.28	$164.30	$165.19	$5.34	$-52.94	False
2023-09-27	2023-10-02	Short	Long	44	$120.20	$115.63	$201.08	$171.04	$166.54	$-27.00	$174.08	True
2023-10-04	2023-10-18	Long	Short	48	$111.50	$112.95	$69.60	$163.04	$168.92	$-35.28	$34.32	True
2023-11-08	2023-11-17	Long	Short	52	$102.93	$104.96	$105.56	$142.40	$144.46	$-14.42	$91.14	True
2023-12-04	2023-12-18	Long	Short	53	$102.43	$101.65	$-41.34	$144.87	$149.68	$-33.67	$-75.01	False
2024-01-03	2024-01-08	Short	Long	52	$103.22	$100.92	$119.60	$152.33	$149.50	$-16.98	$102.62	True
2024-01-11	2024-01-24	Long	Short	55	$98.67	$99.60	$51.15	$145.28	$144.92	$2.52	$53.67	True
2024-01-30	2024-02-02	Short	Long	52	$104.85	$101.97	$149.76	$150.12	$152.24	$14.84	$164.60	True
2024-03-01	2024-04-17	Short	Long	52	$105.84	$118.63	$-665.08	$152.81	$156.40	$25.13	$-639.95	False
2024-04-26	2024-05-09	Long	Short	44	$117.96	$118.44	$21.12	$165.89	$165.45	$2.64	$23.76	True
2024-05-23	2024-05-31	Long	Short	46	$113.51	$117.26	$172.50	$156.94	$162.30	$-32.16	$140.34	True
2024-06-14	2024-06-24	Long	Short	49	$109.11	$114.05	$242.06	$152.57	$159.31	$-40.44	$201.62	True
2024-07-17	2024-07-22	Short	Long	46	$117.64	$115.27	$109.02	$161.23	$156.99	$-25.44	$83.58	True
2024-08-08	2024-08-20	Short	Long	46	$117.89	$114.58	$152.26	$144.49	$144.69	$1.40	$153.66	True
2024-09-10	2024-09-17	Long	Short	50	$110.82	$114.18	$168.00	$138.23	$143.41	$-36.26	$131.74	True
2024-10-01	2024-10-15	Short	Long	47	$119.93	$120.35	$-19.74	$149.70	$147.73	$-13.79	$-33.53	False
2024-10-29	2024-11-06	Long	Short	47	$117.28	$121.00	$174.84	$148.52	$157.72	$-64.40	$110.44	True
2024-11-26	2024-12-31	Long	Short	48	$117.97	$107.57	$-499.20	$162.53	$144.84	$106.14	$-393.06	False
2025-01-10	2025-01-13	Long	Short	51	$106.54	$109.29	$140.25	$153.14	$155.35	$-13.26	$126.99	True
2025-01-15	2025-01-22	Short	Long	49	$111.51	$109.53	$97.02	$158.33	$156.41	$-11.52	$85.50	True
2025-02-12	2025-02-18	Long	Short	52	$107.35	$110.14	$145.08	$154.90	$156.16	$-8.82	$136.26	True
2025-02-20	2025-02-25	Short	Long	50	$112.00	$109.73	$113.50	$158.79	$156.42	$-14.22	$99.28	True
2025-03-05	2025-03-07	Long	Short	54	$105.44	$109.02	$193.32	$150.81	$156.34	$-38.71	$154.61	True
2025-03-17	2025-04-03	Short	Long	50	$113.76	$112.43	$66.50	$158.72	$156.12	$-18.20	$48.30	True

The trade blotter above shows all transactions executed during our testing period, including entry/exit dates, positions, Z-scores, and P&L for each trade. This record demonstrates our strategy’s effectiveness in capturing price divergences between XOM and CVX.

	timestamp	Equity	Daily Return	Drawdown	Trade Action
0	2022-07-27	10000.00	0.00%	0.00%	ENTRY: Short
1	2022-08-03	10116.76	2.78%	0.00%	EXIT: $66.30 (PROFIT)
2	2022-08-23	10116.76	0.00%	0.00%	ENTRY: Short
3	2022-09-01	10456.07	1.49%	0.00%	EXIT: $171.93 (PROFIT)
4	2022-09-20	10456.07	-0.00%	0.00%	ENTRY: Long
...	...	...	...	...	...
73	2025-02-25	12883.54	1.20%	0.00%	EXIT: $99.28 (PROFIT)
74	2025-03-05	12883.54	-0.00%	0.00%	ENTRY: Long
75	2025-03-07	13228.16	0.87%	0.00%	EXIT: $154.61 (PROFIT)
76	2025-03-17	13228.16	0.00%	0.00%	ENTRY: Short
77	2025-04-03	13311.26	4.05%	0.00%	EXIT: $48.30 (PROFIT)

78 rows × 5 columns

The ledger results track daily equity changes and performance throughout the backtest period. This confirms the strategy’s stability across various market conditions, with strong results during periods of heightened volatility.

Key observations:

1. Position Management: Consistent implementation of the hedge ratio maintained market neutrality.

2. Holding Periods: Average holding period of 14.4 days, shorter than our calculated half-life (42.41 days).

3. Equity Growth: Steady equity progression with manageable drawdowns, confirming the strategy’s robust risk-return profile.

Key Performance Metrics

Profitability

• Total Profit: $1,665.63
• Average Profit per Trade: $41.64
• Profit Factor: 1.93

Trade Statistics

• Win Rate: 77.50%
• Average Win: $111.25
• Average Loss: -$198.13
• Average Holding Period: 11.75 days

Risk Metrics

• Maximum Drawdown: -13.38%
• Sharpe Ratio: 0.79
• Sortino Ratio: 0.90

Overall Result

• Initial Capital: $10,000
• Final Equity: $13,632.27
• Return: 36.32%

Drawdown Analysis

The drawdown chart above shows the percentage decline from peak equity values throughout the testing period. The maximum drawdown of 13.38% is moderate and reflects the strategy’s effective risk management, especially considering the volatile nature of energy stocks.

Strategy Insights

Several key observations emerge from our trading results:

1. High Win Rate: The 77.50% win rate confirms the strong mean-reversion tendency between XOM and CVX, validating our statistical approach.

2. Asymmetric Risk-Reward: While average losses (-$198.13) are larger than average wins ($111.25), the high win rate more than compensates, resulting in a strong profit factor of 1.93.

3. Reasonable Drawdowns: The maximum drawdown of 13.38% is well within acceptable limits for a market-neutral strategy, indicating effective risk management.

4. Efficient Holding Period: Despite the relatively long half-life estimate (42.41 days), actual trades averaged just 11.75 days, suggesting that mean reversion often occurs faster than the statistical estimate.

5. Risk-Adjusted Performance: The Sharpe ratio (0.79) and particularly the Sortino ratio (0.90) demonstrate good risk-adjusted returns, with especially strong performance during downside market moves.

Challenges and Considerations

While the overall performance is strong, several factors warrant attention:

1. Correlation Stability: The negative stability score (-21.716) indicates significant variations in the correlation between XOM and CVX over time, which could introduce periods of strategy underperformance.

2. Long Half-Life: The relatively long statistical half-life (42.41 days) suggests that some divergences might persist for extended periods, requiring patience and strong risk management.

3. VIX Adjustment: The strategy was implemented without VIX-based volatility adjustments. Incorporating market volatility indicators could potentially enhance performance by adjusting position sizing during high-volatility periods.

4. Market Regime Sensitivity: Energy stocks can be subject to distinct market regimes (contango vs. backwardation in oil futures, for example) which may temporarily affect the pair relationship.

Conclusion

The XOM/CVX pair demonstrates strong characteristics for statistical arbitrage trading, with solid statistical validation and impressive backtest performance. The high win rate, favorable profit factor, and moderate drawdowns all suggest this pair offers a compelling opportunity for market-neutral trading.

Despite some concerns regarding correlation stability and the relatively long half-life, the actual trading results show that the strategy effectively captures profits from temporary price divergences while maintaining low correlation to broader market movements.

Based on these results, the XOM/CVX pair represents a prime example of statistical arbitrage potential in the energy sector, offering both strong returns and diversification benefits within a broader investment portfolio.

Implementation Recommendation

For optimal implementation, we recommend:

1. Using the established hedge ratio of 0.1816 to maintain market neutrality
2. Monitoring correlation stability and adjusting position sizing accordingly
3. Implementing a trailing stop-loss strategy during periods of extended divergence
4. Considering additional filters based on energy sector fundamentals